How about those that don't believe?

“So what happens to the five billion or so non-Christians? Where do they go?”

Well, there are more than five billion non-Christians to consider, since many non-Christians already have lived and died, and more are yet to be born who surely will not believe.

First, you must delineate the time before the start of the New Covenant, and the time since. With the New Covenant, God provided a new path to salvation, through Christ. Before that time, the path to salvation was not quite the same.

Under the Old Covenant, Jews obtained salvation not through the Law, but through faith in God. However, obedience to the Law was confirmation of that faith. Those who professed the belief, but did not follow the law did not have true faith. For Gentiles, who did not have the Law, the situation is explained by Paul in Romans 2, verses 12-15:
“ 12For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law;
13(For not the hearers of the law are just before God, but the doers of the law shall be justified.
14For when the Gentiles, which have not the law, do by nature the things contained in the law, these, having not the law, are a law unto themselves:
15Which shew the work of the law written in their hearts, their conscience also bearing witness, and their thoughts the mean while accusing or else excusing one another;)”

So, this demonstrates that God would judge those who did not have the Law by the witness of their conscience. If they, by nature, lived righteously (according to the law written in their hearts), their consciences would acquit them, and if they lived sinfully, their consciences would accuse them.

Nowadays, since we have been given the Gospel of salvation for all peoples, few should be able to appeal to a case such as this. If you seek the Lord today, it is likely that you will be able to find Him through the Gospel, and so you would be expected to follow the teachings of the Gospel to obtain salvation.

Perhaps there are still a few who the Gospel has not reached, in which case I expect the Lord would judge those without the Gospel in a similar manner as those in past times, who were without the Law.